We show that targets (REF ), (REF ), and (REF ) all lead to a stepsize that satisfies Assumption REF . In addition, we show a useful bound for the \((\ell + 1)\) st gradient component. Analogue proofs, with the same line of thought, can be found in, e.g, [1]} or [2]}. First notice that, with these choices for the target, for \(k\)  sufficiently large:
\(\xi _{\rm {low}}\cdot \alpha _{k}^{\rm {BB1}} \le \alpha _{k}^{\rm {}}(\tau _k) \le \xi ^{\rm {up}}\cdot \lambda _n\) 
